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# BGW2012 - Linear balanceable and subcubic balanceable graphs

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### BGW2012 - Linear balanceable and subcubic balanceable graphs

1. 1. Linear balanceable and subcubic balanceable graphs Théophile Trunck BGW 2012 November 2012Théophile Trunck (BGW 2012) Balanceable graphs November 2012 1 / 23
2. 2. Co-authorsJoint work with: Pierre Aboulker, LIAFA, Paris Marko Radovanović, Union University, Belgrade Nicolas Trotignon, CNRS, LIP, Lyon Kristina Vušković, Union University, Belgrade and Leeds University Théophile Trunck (BGW 2012) Balanceable graphs November 2012 2 / 23
3. 3. MotivationConjecture (Morris, Spiga and Webb)If G is cubic and every induced cycle has length divisible by 4, then G has a pairof twins. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 3 / 23
4. 4. MotivationConjecture (Morris, Spiga and Webb)If G is cubic and every induced cycle has length divisible by 4, then G has a pairof twins. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 3 / 23
5. 5. DeﬁnitionsDeﬁnitionLet G be a bipartite graph, we say that G is balanceable if we can give weights+1, −1 to edges such that the weight of every induced cycle is divisible by 4. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 4 / 23
6. 6. CharacterizationTheorem (Truemper)A bipartite graph is balanceable if and only if it does not contain an odd wheel noran odd 3-path conﬁguration. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 5 / 23
7. 7. ConjectureConjecture (Conforti, Cornuéjols and Vušković)In a balanceable bipartite graph either every edge belongs to some R10 or there isan edge that is not the unique chord of a cycle. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 6 / 23
8. 8. Main resultsTheoremIf G is a 4-hole free balanceable graph on at least two vertices, then G contains atleast two vertices of degree at most 2.TheoremIf G is a cubic balanceable graph that is not R10 , then G has a pair of twins noneof whose neighbors is a cut vertex of G .CorollaryThe conjecture is true if G does not contain a 4-hole or if ∆(G ) ≤ 3. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 7 / 23
9. 9. DecompositionTheorem (Conforti, Cornuéjols, Kappor and Vušković + Conforti and Rao +Yannakakis + easy lemma)Let G be a connected balanceable graph. If G is 4-hole free, then G is basic, or has a 2-join, a 6-join or a star cutset. If ∆(G ) ≤ 3, then G is basic or is R10 , or has a 2-join, a 6-join or a star cutset. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 8 / 23
10. 10. The Good B1 B2 B2 C1 C2 C2 A1 A2 A2 X1 X2 X2 Figure : 2-join Théophile Trunck (BGW 2012) Balanceable graphs November 2012 9 / 23
11. 11. The Bad B1 B2 B2 C1 C2 C2 A1 A2 A2 X1 X2 X2 Figure : 6-join Théophile Trunck (BGW 2012) Balanceable graphs November 2012 10 / 23
12. 12. The UglyDeﬁnitionA star cutset in a graph G is a set S of vertices such that: G S is disconnected. S contains a vertex v adjacent to all other vertices of S.We note (x, R) the star cutset. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 11 / 23
13. 13. In a perfect worldTheoremLet G be bipartite 4-hole free with no-star cutset, then {2, 6}-join blocks preserve: Being balanceable; Having no star cutset; Having no 6-join. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 12 / 23
14. 14. In a perfect worldTheoremLet G be bipartite 4-hole free with no-star cutset, then {2, 6}-join blocks preserve: Being balanceable; Having no star cutset; Having no 6-join.TheoremLet G be a bipartite 4-hole free graph. Let X1 , X2 be a minimally-sided {2, 6}-join.If G has no star cutset, then the block of decomposition G1 has no {2, 6}-join. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 12 / 23
15. 15. Crossing 2-join Théophile Trunck (BGW 2012) Balanceable graphs November 2012 13 / 23
16. 16. Star cutset, againDeﬁnitionA star cutset in a graph G is a set S of vertices such that: G S is disconnected. S contains a vertex v adjacent to all other vertices of S.DeﬁnitionA double star cutset in a graph G is a set S of vertices such that: G S has two disconnected components C1 and C2 . S contains an edge uv such that every vertex in S is adjacent to u or v .We call C1 ∪ S and C2 ∪ S the blocks of decomposition, and we note (u, v , U, V )where U ⊆ N(u) and V ⊆ N(v ) the double star cutset. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 14 / 23
17. 17. Extreme double star cutsetTheoremLet G be a 2-connected 4-hole free bipartite graph that has a star cutset. Let G1be a minimal side of a minimally-sided double star cutset of G . Then G1 does nothave a star cutset. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 15 / 23
18. 18. Extreme double star cutset u v G1 is 2-connected. U V Théophile Trunck (BGW 2012) Balanceable graphs November 2012 16 / 23
19. 19. Extreme double star cutset G1 is 2-connected. (x, R) a star cutset in G1 . x u v |R ∩ S| ≤ 1. If R ∩ {u, v } = ∅ then U V (x, y ∈ R, R {y }, ∅) is a double star cutset in G . Théophile Trunck (BGW 2012) Balanceable graphs November 2012 17 / 23
20. 20. Extreme double star cutset G1 is 2-connected. C component in G1 ({x} ∪ R) x u v with C ∩ ({v } ∪ V ) = ∅. C U =∅ U V (x, u, R {u}, U) is a double star cutset in G . Théophile Trunck (BGW 2012) Balanceable graphs November 2012 18 / 23
21. 21. Extreme double star cutset If a component of G1 ({x} ∪ R) x=u v contains a vertex from U or V , it contains vertex from G1 S. U V (x, v , U ∪ R {v }, V ) is a double star cutset in G . Théophile Trunck (BGW 2012) Balanceable graphs November 2012 19 / 23
22. 22. Extreme double star cutset {v } ∪ V are in the same component in G1 ({x} ∪ R) u v If a component of G1 ({x} ∪ R) contains a vertex from U, it contains vertex from G1 S. x∈U V (x, u, R {u}, U {x}) is a double star cutset in G . Théophile Trunck (BGW 2012) Balanceable graphs November 2012 20 / 23
23. 23. Sketch of the proofTheoremIf G is a 4-hole free balanceable graph on at least two vertices, then G contains atleast two vertices of degree at most 2.Proof. If we have a cut vertex it is easy. Assume there is a star cutset. Take a double star cutset such that the block G has no star cutset. G is basic or has {2, 6}-join. If G is basic ﬁnd two vertices of degree 2. Take (X1 , X2 ) a minimally-sided {2, 6}-join with small intersection with the double star cutset. Now G1 is basic, ﬁnd good vertices in it. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 21 / 23
24. 24. Open questionsQuestionHow to build every cubic graph such that every induced cycle has length divisibleby 4 ?Conjecture (Conforti, Cornuéjols and Vušković)In a balanceable bipartite graph either every edge belongs to some R10 or there isan edge that is not the unique chord of a cycle. Théophile Trunck (BGW 2012) Balanceable graphs November 2012 22 / 23
25. 25. Thanks for you attention.Théophile Trunck (BGW 2012) Balanceable graphs November 2012 23 / 23